Question 1:- Find the zeros of the quadratic polynomial x² + 7x + 12, and verify the relation between the zeros and its coefficients. Solution:- We have, f(x) = x² + 7x + 12 f(x) = x² + 7x + 12 ⇒ f(x) = x² + 4x + 3x + 12 ⇒ f(x) = x(x … Read more
Question:- Two group are competing for the position on the Board of Directors of a corporation. The probabilities that the first and second group will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probabilities of introducing a new product is 0.7 and the corresponding probability is 0.3, if the second … Read more
Question 1:- An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X ? Is X is random variable ? If yes, find the mean of X. Solution:- Let X = represent the number … Read more
Question:- In a certain collage, 4% of boys and 1% of girls are taller than 1.75 metres, Furthermore, 60% of the students in the college are girls. A student is selected at random from the college and is found to be taller than 1.75 metres. Find the probability that the selected student is a girl. … Read more
Question :- Find the value of k for which the pair of equations kx + 2y = 3 and 3x + 6y = 10 has a unique solution. Solution:- Given pair of linear equations is kx + 2y = 3 …….(i) 3x + 6y = 10 …….(ii) $a_1 = k,b_1 = 2, c_1 = 3$ … Read more
Question:- Prove that root n is not a rational number, if n is not perfect square Solution:- Let √n be a rational number. ∴ , where p and q are co-prime integers q≠ 0. On squaring both side we get ….(i) ⇒ n divides p² ⇒ n divides p … (ii) [If n divides p², … Read more
Question :- Prove that √2 + √5 is irrational Solution:- Let us suppose that √2 + √5 is rational. Let , where a and b coprime integers and b ≠ 0. Now Squaring both side ⇒ √2 is rational, since is rational. This contradict the fact that √2 is an irrational number. So, our assumption … Read more
Question:- Prove that √p + √q is irrational, where p and q are primes. Solution:- Let us suppose that √p + √q is rational. Let √p + √q = a where a is rational number. ⇒ √p = a – √q On squaring both side we get ⇒ [Using formula ] The above statement is a … Read more
Question:- Prove that 2 – √ 3 is irrational, given that √3 is irrational Solution: Let 2 – √3 be a rational number. Then there exist positive integers ‘a’ and ‘ b’, where a and b are integers having no common factor other than 1 and b ≠ 0 such that ⇒ Since a and … Read more
Question:- Prove that root 5 is an irrational number Solution:- Let us assume that on the contrary that √5 is a rational number. Then there exist positive integers ‘a’ and ‘ b’ such that ⇒ Where and are coprime i.e. there HCF is 1 Squaring both side ⇒ ⇒ is divisible by 5 then $a$ … Read more